Elliptic threefolds with high Mordell-Weil rank
type: Item
instance of: scholarly article
title: Elliptic threefolds with high Mordell-Weil rank
author: Antonella Grassi
author: Timo Weigand
published in: Communications in Number Theory and Physics
publication date: 2022-10-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/2105.02863
review text: The authors present examples of smooth Calabi-Yau \(3\)-folds with the Mordell-Weil (MW) rank \(10\), the highest currently known value, and give detailed results on their properties as models of quantum gravity, such as massless particle spectrum and anomaly cancellation conditions. The MW rank determines the rank of the Abelian gauge algebra in \(F\)-theory compactifications on these \(3\)-folds. The \(3\)-folds \(X_i\) studied in the paper are resolutions of \(\bar{X}:=B\times_{\mathbb{P}^1}B'\), with \(B\), \(B'\) rational elliptic surfaces, first considered by Schoen. The ones with MW rank \(10\) specifically were studied by Namikawa and Rossi for their deformation properties and are known as Namikawa examples. The authors prove that the discriminant of the elliptic fibration \(X_i\to B\) is supported on six cuspidal curves with generic fibers of Kodaira type II that upgrade to Kodaira type IV at the six cusps. It is conjectured that the Namikawa examples have the highest MW rank among the Schoen \(3\)-folds. The paper then develops a detailed geometry-physics dictionary for the Namikawa examples. To compute the massless spectrum and the \(U(1)\) charges, the authors explicitly determine, using extensive calculations, the Poincare pairing between \(H^2\) and \(H_2\), the Shioda map, the height pairings, the Gopakumar-Vafa invariants, and some other invariants. It turns out that charged singlet matter does not have to reside at enhancement loci of type \(I_2\), as previously believed. Geometric counterparts of the gravitational and \(U(1)\) anomaly equations are also derived, and it is verified that the anomaly cancellation conditions are satisfied. Finally, the authors obtain birational invariants of non-\(\mathbb{Q}\)-factorial singularities of the Weierstrass model \(\bar{X}\to B\), and compare the latter to the Weierstrass model \(W_{NDE}\to B\) for \(B=\mathbb{P}^2\) constructed by Elkies. In particular, they explicitly describe a log canonical model \(W_{\…
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 14D99
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 81T30
zbMATH DE Number: 7606767
zbMATH Keywords: Calabi-Yau 3-folds
zbMATH Keywords: Mordell-Weil rank
zbMATH Keywords: rational elliptic surfaces
zbMATH Keywords: elliptic fibration
zbMATH Keywords: Schoen 3-folds
zbMATH Keywords: Namikawa examples
zbMATH Keywords: quantum gravity
zbMATH Keywords: abelian gauge algebra
zbMATH Keywords: F-theory compactification
zbMATH Keywords: massless particle spectrum
zbMATH Keywords: U(1) charge
zbMATH Keywords: charged singlet matter
zbMATH Keywords: anomaly cancellation
zbMATH Keywords: Weierstrass model
zbMATH Keywords: Elkies models
zbMATH Keywords: birational Calabi-Yau
MaRDI profile type: Publication
DOI: 10.4310/CNTP.2022.V16.N4.A3
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description: scientific article; zbMATH DE number 7606767